A hemispherical shell of radius R is capped on a vertical hollow cone of base radius R and height 4 3 R (as in an ice cream cone). Both the cone and the shell have the same mass per unit area. Find

Question

A hemispherical shell of radius R is capped on a vertical hollow cone of base radius R and height 4 3 R ​ (as in an ice cream cone). Both the cone and the shell have the same mass per unit area. Find the distance of the center of mass of the system from the vertex of the cone.

TestHub · Accepted Answer

Shell: m 1 ​ = σ ( 2 π R 2 ) , its COM is at x 1 ​ = 4 3 R ​ + 2 R ​ = 4 5 R ​ from the vertex. Cone: ℓ = R 2 + ( 4 3 R ​ ) 2 ​ = 4 5 R ​ , so m 2 ​ = σπ R ℓ = 4 5 ​ πσ R 2 ; hollow-cone COM at x 2 ​ = 3 2 ​ ⋅ 4 3 R ​ = 2 R ​ from the vertex. Combined: x = m 1 ​ + m 2 ​ m 1 ​ x 1 ​ + m 2 ​ x 2 ​ ​ = 2 + 4 5 ​ 2 ⋅ 4 5 ​ + 4 5 ​ ⋅ 2 1 ​ ​ R = 13/4 25/8 ​ R . So x = 26 25 R ​ . Hence option (D).

Physics - Center of Mass Question with Solution | TestHub

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JEE Main 2028 (Class 11)